    Discontinuous Galerkin Finite Element Methods for 1D Rosenau Equation

In this paper, discontinuous Galerkin finite element methods are applied to one dimensional Rosenau equation. Theoretical results including consistency, a priori bounds and optimal error estimates are established for both semidiscrete and fully discrete schemes. Numerical experiments are performed to validate the theoretical results. The decay estimates are verified numerically for the Rosenau equation.

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1 Introduction

Consider the one dimensional Rosenau equation:

 ut+uxxxxt=f(u)x,(x,t)∈(a,b)×(0,T] (1.1)

with initial condition

 u(x,0)=u0(x),x∈(a,b), (1.2)

and the boundary conditions

 u(a,t) = u(b,t)=0, ux(a,t) = ux(b,t)=0, (1.3)

where is a nonlinear term in of the type , here is a real constant and is a positive integer.

The Rosenau equation is an example of a nonlinear partial differential equation, which governs the dynamics of dense discrete systems and models wave propagation in nonlinear dispersive media.

Recently, several numerical techniques like conforming finite element methods, mixed finite element methods, orthogonal cubic spline collocation methods, etc., were proposed to find the approximate solution of Rosenau equation. The different conforming finite element techniques are used to approximate the solution of Rosenau equation needs -interelement continuity condition and mixed finite element formulations are required -continuity condition. In this article discontinuous Galerkin finite element methods are used to approximate the solution.

The well-posedness results of (1.1)-(1.3) was proved by Park  and Atouani et al. in . Earlier, some numerical methods were proposed to solve the Rosenau equation (1.1)-(1.3) using finite difference methods by Chung , conservative difference schemes by Hu and Zheng  and Atouni and Omrani . Finite element Galerkin method was used by [2, 7], a second order splitting combined with orthogonal cubic spline collocation method was used by Manickam et al.  and Chung and Pani in  constructed a -conforming finite element method for the Rosenau equation (1.1)-(1.3) in two-space dimensions.

In recent years, there has been a growing interest in discontinuous Galerkin finite element methods because of their flexibility in approximating globally rough solutions and their potential for error control and mesh adaptation.

Recently, a cGdG method was proposed by Choo. et. al in . A subdomain finite element method using sextic b-spline was proposed by Battal and Turgut in . But constructing finite elements for fourth order problems becomes expensive and hence discontinuous Galerkin finite element methods can be used to solve fourth order problems .

In this paper, we introduce discontinuous Galerkin finite element methods (DGFEM) in space to solve the one dimensional Rosenau equation (1.1)-(1.3). Comparitive to existing methods our proposed method require less regularity.

The outline of the paper is as follows. In Section 2, we derive the discontinuous weak formulation of the Rosenau equation. In Section 3, we discuss the a priori bounds and optimal error estimates for the semidiscrete problem. In Section 4, we discretize the semidiscrete problem in the temporal direction using a backward Euler method and discuss the a priori bounds and optimal error estimates. Finally, in Section 5, we present some numerical results to validate the theoretical results.

Throughout this paper, denotes a generic positive constant which is independent of the discretization parameter which may have different values at different places.

2 Weak Formulation

In this section, we derive the weak formulation for the problem (1.1)-(1.3).

We discretize the domain into subintervals as

 a=x0

and for . We denote this partition by consisting of sub-intervals . Below, we define the broken Sobolev space and corresponding norm

 Hs(Eh)={v∈L2(a,b):v|In∈Hs(In),∀In∈Eh}

and

 |||v|||Hs(Eh)=⎛⎝∑K∈Eh∥v∥2Hs(K)⎞⎠12.

Now define the jump and average of across the nodes as follows. The jump of a function value across the inter-element node , shared by and denoted by and defined by

 [v(xn)]=v(x−n)−v(x+n).

At the boundary and , we set

 [v(x0)]=−v(x0),and[v(xN)]=v(xN).

The average of a function value across the inter-element node , shared by and denoted by and defined by

 {v(xn)}=12(v(x−n)+v(x+n)).

At the boundary and , we set

 {v(x0)}=v(x0),and{v(xN)}=v(xN).

We multiply (1.1) with and integrate over to obtain

 ∫xn+1xn(ut+uxxxxt)vdx=∫xn+1xnf(u)xvdx. (2.4)

Now, using integration by parts twice in (2.4), we arrive at

 ∫xn+1xnutvdx+∫xn+1xnuxxtvxxdx+uxxxt(xn+1)v(x−n+1)−uxxxt(xn)v(x+n) −uxxt(xn+1)vx(x−n+1)+uxxt(xn)vx(x+n)=∫xn+1xnf(u)xvdx.

Summing over all the elements and using

 ps−qr=12(p+q)(s−r)+12(r+s)(p−q),p,q,rands∈R,

we obtain

 N−1∑n=0∫xn+1xnutvdx+N−1∑n=0∫xn+1xnuxxtvxxdx + (2.5) = N−1∑n=0∫xn+1xnf(u)xvdx.

Since is assumed to be sufficiently smooth, we have . Using this, we write as

 (2.6)

The right hand side of (2.6) was found out using the boundary conditions 1.3. Adding (2.6) to (2.5) we obtain

 N−1∑n=0∫xn+1xnutvdx + (2.7) − N∑n=0{uxxt(xn)}[vx(xn)]−N∑n=0{vxx(xn)}[uxt(xn)]+σ0hβN∑n=0[v(xn)][ut(xn)] +

We define the bilinear form as

 B(u,v) = A(u,v)+Jσ0(u,v)+Jσ1(u,v), (2.8)

where

 A(u,v) = N−1∑n=0∫xn+1xnuxxvxxdx+N∑n=0({uxxx(xn)}[v(xn)]+{vxxx(xn)}[u(xn)]) − N∑n=0({uxx(xn)}[vx(xn)]+{vxx(xn)}[ux(xn)]),

and

 Jσ0(u,v)=N∑n=0σ0hβ[u(xn)][v(xn)],Jσ1(u,v)=N∑n=0σ1h[ux(xn)][vx(xn)].

In (2.8), and are the penalty terms and . The value of will be defined later.

The weak formulation of (1.1)-(1.3) as follows: Find , such that

 (ut,v)+B(ut,v) = (f(u)x,v),∀v∈Hs(Eh),t>0 (2.9) u(x,0) = u0(x). (2.10)

Below, we state and prove the consistency result of the weak formulation (2.9)-(2.10).

Theorem 2.1.

Let be a solution of the continuous problem (1.1)-(1.3). Then satisfies the weak formulation (2.9)-(2.10). Conversely, if for is a solution of (2.9)-(2.10), then satisfies (1.1)-(1.3).

Proof.

Let and . Multiply (1.1) by and integrate from to . Sum over all and using (2.5), (2.6) and (2.7), we obtain the weak formulation (2.9).

Conversely, let and , the space of infinitely differentiable functions with compact support in . Then, (2.9) becomes

 ∫xn+1xnutvdx+∫xn+1xnuxxtvxxdx=∫xn+1xnf(u)xvdx. (2.11)

Applying integration by parts twice on the second term on the left hand side of (2.11) to obtain,

 ∫xn+1xnuxxtvxxdx=∫xn+1xnuxxxxtvdx,

as is compactly supported on . This immediately yields

 ut+uxxxxt=f(u)x,a.e inIn. (2.12)

Consider the node shared between and . Choose , multiply (2.12) by and integrate over to obtain

 ∫Ik−1∪Ikutvdx+∫Ik−1∪Ikuxxxxtvdx=∫Ik−1∪Ikf(u)xvdx. (2.13)

Applying integration by parts twice on the second term of (2.13) and using , we obtain

 ∫Ik−1∪Ikutvdx + ∫Ik−1∪Ikuxxtvxxdx+[uxxxt(xk)]v(xk)−[uxxt(xk)]vx(xk) (2.14) = ∫Ik−1∪Ikf(u)xvdx.

On the other hand, we have from (2.9) for the choice of and ,

 ∫Ik−1∪Ikutvdx+∫Ik−1∪Ikuxxtvxxdx=∫Ik−1∪Ikf(u)xvdx. (2.15)

Comparing (2.14) and (2.15) and using the fact that is arbitrary, we obtain

 [uxxxt(xk)]=0.

Thus and hence, from (2.13), we obtain

 ut+uxxxxt=f(u)xa.e in (a,b). (2.16)

This completes the proof. ∎

3 Semidiscrete DGFEM

In this section, we discuss the a priori bounds and optimal error estimates for the semidiscrete Galerkin method.

We define a finite dimensional subspace of as

 Dk(Eh)={v∈L2(a,b):v|In∈Pk(In),In∈Eh}.

The weak formulation for the semidiscrete Galerkin method is to find such that

 (uht,χ)+B(uht,χ) = (f(uh)x,χ),for allχ∈Dk(Eh), (3.1) uh(0) = uh0, (3.2)

where is an appropriate approximation of which will be defined later.

3.1 A priori Bounds

In this sub-section, we derive the a priori bounds.

Define the energy norm

 ||u||2E=N∑n=0∫xn+1xnu2xx+N∑n=0σ0hβ|[u(xn)]|2+N∑n=0σ1h|[ux(xn)]|2.

We note from  that is coercive with respect to the energy norm, i.e.,

 B(v,v)≥C||v||2E,v∈Dk(Eh),

for sufficiently large values of and .

Observe that (3.1

) yields a system of non-linear ordinary differential equations and the existence and uniqueness of the solution can be guaranteed locally using the Picard’s theorem. To obtain existence and uniqueness globally, we use continuation arguments and hence we need the following

a priori bounds.

Theorem 3.1.

Let be a solution to (3.1) and assume that is bounded. Then there exists a positive constant such that

 ∥uh(t)∥+∥uh(t)∥E≤C(∥uh0∥E). (3.3)
Proof.

On setting in (3.1), we obtain

 (uht,uh)+B(uht,uh)=(f(uh)x,uh). (3.4)

We rewrite the equation (3.4) as

 12ddt∥uh(t)∥2+12ddtB(uh(t),uh(t))=(f′(uh)uhx,uh).

Integrating from to , we obtain

 ∥uh(t)∥2+B(uh(t),uh(t))=∥uh(0)∥2+B(uh(0),uh(0))+∫t0(f′(uh)uhx,uh)ds. (3.5)

On using the coercivity of and the boundedness of , we arrive at

 ∥uh(t)∥2+C∥uh(t)∥2E≤∥uh(0)∥2+B(uh(0),uh(0))+C∫t0(uhx,uh)ds. (3.6)

Using the Cauchy Schwarz inequality and the Poincaré inequality on the right hand side of (3.6), we obtain

 (3.7)

An application of Gronwall’s inequality yields the desired a priori bound for . ∎

3.2 Error Estimates in the energy and L2-norm

In this subsection, we derive the optimal error estimates in energy and -norm.

Often a direct comparison between and does not yield optimal rate of convergence. Therefore, there is a need to introduce an appropriate auxiliary or intermediate function so that the optimal estimate of is easy to obtain and the comparision between and yields a sharper estimate which leads to optimal rate of convergence for . In literature, Wheeler  for the first time introduced this technique in the context of parabolic problem. Following Wheeler , we introduce be an auxiliary projection of defined by

 B(u−~u,χ)=0,for allχ∈Dk(Eh). (3.8)

Now set the error and split as follows: , where and . Below, we state some error estimates for and its temporal derivative.

Lemma 3.1.

For and then there exists a positive constant independent of such that the following error estimates for hold:

 ∥∥∥∂lη∂tl∥∥∥E ≤ Chmin(k+1,s)−2(l∑m=0|||∂mu∂tm|||Hs(Eh)), ∥∥∥∂lη∂tl∥∥∥ ≤ Chmin(k+1,s)(l∑m=0|||∂mu∂tm|||),l=0,1.
Proof.

We split as follows:

 η=u−~u=(u−¯u)−(~u−¯u)=ρ−ξ,

where , and

is an interpolant of

satisfying good approximation properties. Now from (3.8), we have

 B(ξ,χ)=B(ρ,χ). (3.9)

We note that satisfies the following approximation property :

 ∥ρ∥Hq(In)≤Chmin(k+1,s)−q∥u∥Hs(In).

Set in (3.9) to obtain

 B(ξ,ξ)=B(ρ,ξ).

A use of coercivity of and the assumption that is a sufficiently smooth interpolant of , we obtain

 C∥ξ∥2E≤N−1∑n=0∫xn+1xnρxxξxxdx+N∑n=0{ρxxx(xn)}[ξ(xn)]−N∑n=0{ρxx(xn)}[ξx(xn)]. (3.10)

Now we estimate the first term as follows:

 N−1∑n=0∫xn+1xnρxxξxxdx ≤ |||ρxx||||||ξxx|||≤16C∥ρ∥2H2+C6∥ξ∥2E, (3.11) ≤ Ch2min(k+1,s)−4|||u|||2Hs(Eh)+C6∥ξ∥2E.

Estimating the second term using Hölder’s inequality, trace inequality and the Young’s inequality, we obtain

 N∑n=0{ρxxx(xn)}[ξ(xn)]≤Ch2min(k+1,s)−6+β−1|||u|||2Hs(Eh)+C6∥ξ∥2E. (3.12)

Similarly the last term can be estimated as

 N∑n=0{ρxx(xn)}[ξx(xn)] ≤ Ch2min(k+1,s)−6+β−1|||u|||2Hs(Eh)+C6∥ξ∥2E. (3.13)

Combining (3.11)-(3.13), we obtain the following bound for when

 ∥ξ∥E≤Chmin(k+1,s)−2|||u|||Hs(Eh). (3.14)

Now using , we obtain the energy norm estimate for . For the -estimate of , we use the Aubin Nitsché duality argument. Consider the dual problem

 d4ϕdx4=η,x∈(a,b), ϕ(a)=ϕ(b)=0, ϕ′(a)=ϕ′(b)=0.

We note that satisfies the regularity condition . Consider

 (η,η)=(η,ϕxxxx)=N−1∑n=0∫xn+1xnϕxxηxxdx+N∑n=0{ϕxxx(xn)}[η(xn)]−N∑n=0{ϕxx(xn)}[ηx(xn)].

Since , we can write

 ∥η∥2=(η,η)−B(η,~ϕ) = N−1∑n=0∫xn+1xn(ϕ−~ϕ)xxηxxdx+N∑n=0{(ϕ−~ϕ)xxx(xn)}[η(xn)] (3.15) − N∑n=0{(ϕ−~ϕ)xx(xn)}[ηx(xn)],

where is a continuous interpolant of and satisfies the approximation property:

 ∥ϕ−~ϕ∥Hq≤Chs−q∥ϕ∥Hs. (3.16)

We use the approximation property (3.16), the energy norm estimate for and the regularity result to bound each term on the right hand side of (3.15) and obtain the estimate for as:

 ∥η∥≤Chmin(k+1,s)|||u|||Hs(Eh).

For the estimates of the temporal derivative of , we differentiate (3.8) with respect to and repeat the arguments. Hence, it completes the rest of the proof. ∎

The following Lemma is useful to prove the error estimates:

Lemma 3.2.

Let where . Then there exists a positive constant independent of such that,

 Ch−4n∥v∥2L2(In)≤∥vxx∥2L2(In),

where .

Proof.

We define the reference element as

 ^In={(1hn)x,x∈In}.

Since , we have the following relation (refer ) for the norms in the reference element and the interval

 |^v|Hr(^In)=hr−d2n|v|Hr(In).

In one space dimension, i.e., , we have

 ∥^v∥L2(^In)=h−12n∥v∥L2(In),and|^v|H2(^In)=h32n|v|H2(In). (3.17)

By the equivalence of norm (refer ), we have

 C0∥^v∥L2(^In)≤∥^vxx∥L2(^In)≤C1∥^v∥L2(^In). (3.18)

Now from (3.17) and (3.18), we obtain

 C0h−12n∥v∥L2(In)≤h32n∥vxx∥L2(In).

Rearranging the terms and squaring on both sides, we obtain the desired estimate. ∎

To obtain the error estimates, we subtract (3.1) from (2.9) and using the auxiliary projection (3.8), we obtain the following error equation

 (θt,χ)+B(θt,χ)=(ηt,χ)+(f(uh)x−f(u)x,χ). (3.19)

Now we state and prove the following theorem.

Theorem 3.2.

Let and be the solutions of (3.1) and (2.9), respectively. Let be the elliptic projection of , i.e., . Then for and there exists a positive constant independent of such that

 ∥u(t)−uh(t)∥E ≤ Chmin(k+1,s)−2∥u∥H1(0,T;Hs(Eh)), ∥u(t)−uh(t)∥ ≤ Chmin(k+1,s)∥u∥H1(0,T;Hs(Eh)).
Proof.

Setting in (3.19), we obtain

 (θt,θ)+B(θt,θ)=(ηt,θ)+(f(uh)x−f(u)x,θ). (3.20)

Now, we write equation (3.20) as

 12ddt(∥θ(t)∥2+B(θ(t),θ(t)))=(ηt,θ)+(f(uh)x−f(u)x,θ).

Integrating with respect to from to and noting that , we obtain

 ∥θ(t)∥2+B(θ(t),θ(t))=∫T0(ηt,θ)ds+∫T0(f(uh)x−f(u)x,θ)ds. (3.21)

We use integration by parts on the nonlinear term to obtain,

 ((f(uh)−f(u))x,θ)=N∑n=0{f(uh)−f(u)}[θ(xn)]+[f(uh)−f(u)]{θ(xn)}+(f(u)−f(uh),θx). (3.22)

Using the Cauchy Schwarz’s and Young’s inequality, we bound the last term of (3.22) as

 (f(u)−f(uh),θx)≤C(∥η∥2+∥θ∥2+∥θ∥2E).

Now for the first term in (3.22), we use Hölder’s inequality to write

 N∑n=0{f(uh)−f(u)}[θ(xn)]≤(N∑n=0|{f(uh)−f(u)}|2)1/2(N∑n=0|[θ(xn)]|2)1/2. (3.23)

As earlier in (3.12), we use the penalty term to write (3.23) as

 N∑n=0{f(uh)−f(u)}[θ(xn)]≤Chβ−12∥e∥∥θ∥E≤C∥e∥∥θ∥E,sinceβ≥3.

A similar bound for the second term can be obtained as follows. Using the Hölder’s inequality, we write

 N∑n=0[f(uh)−f(u)]{θ(xn)}≤(N∑n=0hβ|[f(uh)−f(u)]|2)1/2(N∑n=0h−β|{θ(xn)}|2)1/2. (3.24)

Using the trace inequality, we obtain

 N∑n=0[f(uh)−f(u)]{θ(xn)} ≤ (Chβ−12∥e∥)(N∑n=0h−β−1∥θ∥2L2(In